Ex: Toán học - Lớp 8 - 2024-11-27 20:49:31

The problem presented involves a triangle \( ABC \) with specific geometric properties and asks for proofs and calculations related to triangle similarity and lengths. Here’s a step-by-step breakdown of the tasks:

### a) Prove that \(\frac{AH}{HC} = \frac{HM}{CM}\)

1. **Identify Triangles**: Consider triangles \( AHC \) and \( HMC \).
2. **Use Similarity**: If angle \( A \) and angle \( M \) are the same (since \( H \) is on line segment \( AC \)), the triangles \( AHC \) and \( HMC \) can be shown to be similar due to AA criterion (Angle-Angle).
3. **Set up Ratios**: From similarity, we get the ratio of corresponding sides, leading to the relation \(\frac{AH}{HC} = \frac{HM}{CM}\).

### b) Prove that \( AK \) is perpendicular to \( BM \)

1. **Location of Points**: Since \( K \) is the midpoint of \( HM \), you can use coordinate geometry or properties of medians.
2. **Use Right Triangles**: Utilize the fact that \( AH \) is an altitude, which creates right angles at points.
3. **Show Perpendicularity**: Use slopes or distances to prove \( AK \perp BM \).

### c) Given \( AI = 5 \) cm, \( HI = 4 \) cm, find the length of side \( BC \)

1. **Use Pythagorean Theorem**: In triangle \( AHI \),
\[
AH^2 + HI^2 = AI^2
\]
Substitute the values to find \( AH \).

2. **Find Length \( BC \)**: The length can be determined through ratios established from previous parts (using the similar triangles).

This structured approach will help you prove the claims outlined in the problem. If you need any specific computations or clarifications, let me know!